On a hybrid bike you can fit some of the wider 28mm/narrow 32mm tyres. Try something like a 32mm Panaracer pasela, pumped up to its max pressure, this will cut down on rolling resistance.
Rolling resistance is constant at all speeds, it limits your effiency at low speeds, but above 12-15mph, air resistance becomes the big one. This increases with velocity squared. ie double your speed and the drag increses 4x. On a hybrid bike, small aerobars are probably easier and cheaper to fit then drops. Your frame probably lacks the correct cable mounts for drop bar gear leves. The frame geomety/length may also not be comfortable for use with drops, if it good with your current flats.
Pedal at high cadence, and if you spin out, consider fitting bigger gears. Use toe clips or clipless pedals and stiff cycling shoes.
If you are serious about going fast on the road, then a road bike may be what you are looking for. Some light-touring bikes combine the efficiency of a road bike with the practicality of a touring/hybrid bike, and make excellent all-round bikes.